PNet
Program for the Simulation and Estimation of
Exponential Random Graph (p*) Models
USER MANUAL
Peng Wang
Garry Robins
Philippa Pattison
Department of Psychology
School of Behavioural Science
University of Melbourne
Australia
September, 2009
Table of Content
Introduction ___________________________________________________________ 1
Acknowledgements__________________________________________________________ 1
System Requirements _______________________________________________________ 2
Setup PNet ________________________________________________________________ 2
Update PNet _______________________________________________________________ 2
Using PNet ____________________________________________________________ 3
Start PNet _________________________________________________________________ 3
Simulation _________________________________________________________________ 4
Simulation Setup _________________________________________________________________ 4
Simulation Output ________________________________________________________________ 8
Estimation _________________________________________________________________ 9
Estimation Setup _________________________________________________________________ 9
Estimation Options: _______________________________________________________________ 9
Estimation Output _______________________________________________________________ 11
Goodness of Fit ____________________________________________________________ 12
Goodness of Fit Setup ____________________________________________________________ 12
Goodness of Fit Output ___________________________________________________________ 12
Approximate Bayesian goodness of fit _________________________________________ 13
PNet Extensions _______________________________________________________ 14
BPNet ___________________________________________________________________ 14
Introduction ____________________________________________________________________ 14
Simulation _____________________________________________________________________ 14
Estimation _____________________________________________________________________ 17
Goodness of Fit _________________________________________________________________ 17
References ___________________________________________________________ 18
Appendix A – Sample Files ______________________________________________ 21
Sample Input Files _________________________________________________________ 21
Sample Output Files________________________________________________________ 23
“start_statistics_[session name].txt”, “end_statistics_[session name].txt” and
“sample_statistics_[session name].txt” _______________________________________________ 23
“simulation_[session name].txt” ____________________________________________________ 24
“parameter_[session name].txt” _____________________________________________________ 25
“estimation_[session name].txt” ____________________________________________________ 25
“covariance_[session name].txt” ____________________________________________________ 26
“gof_[session name].txt” __________________________________________________________ 27
Appendix B – Model Parameter Description ________________________________ 32
Non-directed Graphs _____________________________________________________________ 32
Directed Graphs _________________________________________________________________ 34
XPNet Graph Statistics ___________________________________________________________ 36
BPNet Graph Statistics ___________________________________________________________ 39
IPNet Graph Statistics ____________________________________________________________ 42
Introduction
PNet is a program for statistical analysis of exponential random graph (p*)
models (ERGMs). It has three major functionalities:
Simulation:
Simulating network distributions with specified model parameter
values.
Estimation:
Estimating specified ERGM parameters for a given network.
Goodness of Fit:
Testing the goodness of fit of a specified model to a given network
with a particular set of parameters.
Acknowledgements
PNet contains code and ideas from many people. We would like to thank the
following people for contributing to this program.
Carter Butts, Galina Daraganova, Steve M. Goodreau, Mark S. Handcock,
Nicholas Harrigan, David Hunter, Tasuku Igarashi, Johan Koskinen, Dean
Lusher, Martina Morris, Ken Sharpe, Tom A.B. Snijders, Christian E.G. Steglich,
Lei Xing, Yu Zhao.
-1-
System Requirements
Operating
system
Software
Microsoft® Windows operating systems
Microsoft .NET framework version 1.1+
Java TM 2 platform standard edition 5.0+
The Software required is freely available from Microsoft and Sun‟s web site.
Microsoft: www.microsoft.com
Under Download, search for
.NET Framework Version 1.1 Redistributable Package
JAVA TM 2 Platform Standard Edition 6.0
http://java.sun.com/javase/downloads/index.jsp
Setup PNet
PNet consists of two components, a user interface developed in Java “PNet.jar”,
and a simulation/estimation engine “pnet.dll” developed in C to achieve good
performance.
Before installing PNet, make sure you system meet the specified system
requirements as described.
Copy the PNet.jar and pnet.dll into the same folder; you can then start the
program by double clicking on the PNet.jar icon.
Note that PNet.jar and pnet.dll files must be located in the same folder for the
Java interface PNet.jar to call the library functions in pnet.dll.
Update PNet
Newer version of PNet will be available and can be downloaded from
www.sna.unimelb.edu.au/pnet/pnet.html
Please replace your current PNet.jar and pnet.dll files, and update the Java
runtime environment to finish the update.
-2-
Using PNet
To setup a simulation, estimation or goodness of fit, you will need to choose the
relevant options from the user interface and specify several program settings.
The program requires input files, and produces text file output. Samples of input
files and output files can be found in Appendix A.
Start PNet
PNet can be started from the Windows Start menu under Program Files, PNet. At
the top of PNet main window, both Session Name and Session Folder are
required for the output file names and location.
Session Name
Provide a name for the current session for simulation, estimation,
goodness of fit or approximate Bayesian goodness of fit. This name will be
used for the names of the output files. All output files will have file names
that end with the Session Name you provided here, (E.g. if you have a
session name MySession under simulation, you will have an output file
named “simulation-MySession.txt.”)
Session Folder
All program output files will be located in the Session Folder selected here.
You can browse through your system and select the folder by clicking on
the Browse button.
Simulation, Estimation, Goodness of fit and Approximate Bayesian goodness of
fit, each has its own tab, with similar structures. Under each tab, several settings
need to be specified to configure your p* model.
-3-
Simulation
Simulation Setup
To correctly configure simulation, you need to specify several settings
Number of
Actors
Type in the
number of
actors in the
network.
Starting Graph Density
Type in the starting density of a random graph in the simulation that used
to generate the starting simulation network. Type in a floating point
number between 0.0 and 1.0.
Select Network Type:
Models for directed and non-directed networks can be simulated. Choose
the network type here. If you have a model that having constraint on the
maximum number of ties that an actor can have, you should also specify it
here by clicking on the checkbox and type in the maximum degree.
Select Structural Parameters
Click on the
Structural
Parameters
Checkbox to enable
the selection button.
By clicking the
Select Parameters
button, structural
parameter dialog
appears.
Select parameters
for your simulation
model and specify
their values and lambda values if they are higher order parameters.
The “Clear All” button will deselect all parameters and reset their values to
-4-
0. It will also reset lambda to the default value of 2.0
The “Select All” button will select all parameters.
Finish the structural parameter selection by clicking on the OK button.
Select Dyadic Attribute Parameters
Select the
dyadic
attribute
parameters if
you have one or more fixed setting network as network covariates.
By clicking on the Browse button, a file open dialog appears, select the
Covariate network file and click on OK.
The dyadic attribute file is a plain text file having the dyadic attributes
listed in the adjacency matrix format.
-5-
Select Actor Attribute Parameter
Actor attribute Parameters are used in social selection models. You may have
three different types of attributes, Binary, Continuous and Categorical. The
can be selected in a similar manner. The number of attributes should be
specified before select the actual parameters. Attribute files should also be
specified similar to the way how dyadic attribute file is specified. Please check
Appendix A for attribute file format.
Simulation Options
Fix out-degree distribution
Directed networks only, this option will make simulated samples having
identical out-degree distribution.
Fix the graph density
Fix the density of the graph, i.e. the number of arcs/edges in the
network does not change through the entire simulation. Note, as the
number of arcs/edges has been fixed, the arc/edge parameter should
not be selected for simulation.
Structural “0” File:
Structural-zeros refers to the indicators for tie variables that are fixed
-6-
through the simulation. One may fix part of the network by applying a
structural-zero file to the simulation. The file should contain a binary
adjacency matrix with the same number of rows/columns as in the
number of actors. In the matrix, “1” indicates the corresponding tie in
the network is NOT fixed, “0” otherwise.
Please check Appendix A for structural-zero file format.
Pick up Sample
graph Files,
Sample degree
distribution,
Sample
geodesic
distribution and
Sample
clustering
coefficient.
If selected, the
corresponding
samples will set
to be part of the
program output
in separate files.
Burn in
Burn in is the starting period of a simulation during which the network
is evolving and getting adapted to the specified parameter values.
Depends on the size of the network and number of parameter values,
burn-in can vary largely. The larger the network, or the more parameter
involved, the longer burn-in is needed. K-statistics tend to have longer
burn-in.
Number of iterations
Type in the number of iterations after burn in for the simulation
Number of samples to pickup
Type in the number of sample graph statistics should be picked up in
the simulation
-7-
Click on the start button, the simulation starts. PNet will notify you once the
simulation finished.
Simulation Output
File Output
“start_statistics_session.txt”
This file contains the starting graph with selected statistics
“simulation_graph.sps” or “simulation_digraph.sps”
This file contains the SPSS script to plot the scatter-plot and histogram of
the simulated graph statistics using SPSS version 12.0 and above.
“simulation_graph.txt” or “simulation_digraph.txt”
This file contains the list of sample statistics collected during the
simulation.
Using the SPSS script file, you can plot the statistics as scatter-plots and
histograms.
“parameter_graph.txt” or “parameter_digraph.txt”
Showing parameter values used in simulation.
-8-
Estimation
Estimation Setup
To correctly setup an estimation run, several settings need to be configured.
Same as in Simulation, Session Name and Number of Actors should be
provided.
Network File can be selected by clicking on the Browse Button.
Network Type is selected the same way as in Simulation. By setting up the
maximum degree, the model is conditional on the maximum degree of
each actor.
Structural, Covariate and different kind of Actor Attribute parameters can
be selected as in Simulation. See detailed parameter description in
Appendix B.
Starting parameter values can be specified as well at the parameter
selection dialog. If parameter values are not specified, all starting
parameter values are set to 0.0, except the edge or arc parameter which
is calculated based on the density of the network.
Estimation Options:
Fix out-degree distribution
For Directed networks only, this option will estimate conditional models
such that the out-degree distribution will be fixed trough out the estimation.
Fix the graph density
Fix the density of the graph, i.e. the number of arcs/edges in the network
does not change through the entire simulation. Note, as the number of
arcs/edges has been fixed, the arc/edge parameter is not estimable, and it
should not be selected for estimation.
Fix the graph density
By fixing the graph density, the number of arcs/edges will not change
during estimation. Fixing graph density may help convergence for
parameter estimation, especially for large networks.
Note, as the number of arcs/edges has been fixed, the arc/edge
parameter should not be selected for estimation.
-9-
Structural “0” File:
By applying structural “0” file, part of the network under estimation can be
fixed. The file should contain a binary matrix where “1” indicates the
corresponding tie in the network is NOT fixed, “0” otherwise.
Please check Appendix A for the format of the structural-zero file.
Number of Sub-phases
Each sub-phase
refines the
parameter values,
but more subphases do not
guarantee
convergence. The
default value is 5. If
a good set of
starting parameter
values is available,
small number of
sub-phases may
help reduce time
required for the
estimation.
Gaining Factor (a-value)
The a-value is halved after each sub-phase. The default a-value is 0.01.
Smaller a-value may be used, if a good set of starting parameter values is
available.
Multiplication Factor
The larger the multiplication factor, the longer the estimation, but it may
help convergence especially for some large networks. The default value is
10. Set it to the number of parameters may be helpful, and K-statistics
tends to need factor values bigger than 20 (e.g. 20 to 100).
Number of steps in phase 3
In phase 3, the program simulates network graphs using estimated
parameters from phase2, and produce t-statistics according to the
simulation and observation. The default value is 500 steps.
- 10 -
Maximum number of estimation runs
As default, the program will perform 1 run of estimation and quit. Multiple
runs can be performed one after the other; each run uses the parameter
values from the end of the previous run. A better parameter estimate may
be obtained as the new estimation may start with a better set of parameter
values. The program will stop once the model has converged, or the
maximum number of estimation runs has reached.
Do GOF @ model convergence
PNet can perform automatic goodness of fit test once the model under
estimation has converged. The GOF output file will be located in the
session folder.
Update
After first estimation run, the update button will be enabled. It is used
when you want to start next estimation run with previous estimated
parameters so that you may start form a better set of parameters.
Note: PNet will always load the previous estimation session. Please do
NOT use update, if the session name, session folder, or network file has
been changed.
Estimation Output
File Output
“start_statistics_graph.txt”
This file contains the starting graph with graph statistics
“estimation_graph.txt”
Estimation result shows starting parameter values, starting graph statistics
and parameter updates through Phase 2 of the estimation. The final
estimates and estimated covariance matrix are shown at bottom of the file.
“covariance_graph.txt”
It contains the estimated covariance matrix by itself, and it can be used as
the covariance file in Approximate Bayesian goodness of fit.
- 11 -
Goodness of Fit
Goodness of Fit Setup
Most settings for Goodness of Fit is the same as in Simulation, except the
observed network and parameter values are required. The observed network file
can be specified as in Estimation.
Make sure that all parameters are selected; you may do this by using the “select
all button in the parameter selection panel.
The parameter values from your model should also be specified. You can type in
the parameter values as in simulations, or you can use the “Update” button.
Note: Update button will only work once all parameters have been selected (you
may use “select all” button in parameter selection panels). It always loads
parameters from immediate previous estimation session. Please only use update
button immediately after a successful estimation.
Goodness of Fit Output
File Output
“start_statistics_graph.txt”
This file contains the observed graph and graph statistics
“simulation_graph.sps”
This file contains the SPSS script to plot the scatter-plot and histogram of
the simulated graph statistics using SPSS version 15.0 and above.
“accept_graph.txt”
Showing the ratio of accepted simulation tie changes within each
simulation intervals between every two sample graphs.
“gof_graph.txt”
Goodness of fit file contains the original or observed statistics for the given
network graph, and goodness of fit for the specified model for all available
graph statistics.
- 12 -
Approximate Bayesian goodness of fit
In terms of program setup, the difference between approximate Bayesian
goodness of fit and Goodness of fit described in previous section is that
approximate Bayesian goodness of fit requires the estimated covariance matrix
as part of the input.
The covariance file is a text file containing the estimated covariance matrix only.
One may use the covariance file generated from the immediate previous
estimation session; or one can copy the estimated covariance matrix from the
estimation result file and past it into a new text file.
As covariance matrix is only regards to the model estimates, pleas ONLY select
parameters that are included in the model in the parameter selection panels.
Other options for Approximate Bayesian goodness of fit are identical to the
settings in Goodness of fit described in previous section.
- 13 -
PNet Extensions
BPNet
Introduction
BPNet is a program designed for exponential random graph models for bipartite
networks where network ties are only defined between two sets of actors. The
network statistics include both structural and configurations involving actor
attributes.
The general setup and use of the program is similar to PNet. It has a Java user
interface, and C simulation engine. Modifications are made to accompany
features of bipartite networks.
Following are screen shots for BPNet with user instructions aside.
Simulation
The same as in PNet, Session Name and Folder need to be specified first, and
output files will have file names ending with session name, and they will be
located in the session folder.
Numbers of actors (A) and (P) are the number of nodes in set A and set P.
Simulations can be started with a random bipartite networks with specified
density.
Structural parameters can be selected by clicking on the check boxes. The
details of the network configurations can be found in Appendix B.
- 14 -
Parameter values for simulations can be specified during parameter selection.
The default values are 0s.
Actor attribute parameters are selected by click on the check boxes, and type in
the number of attributes for a particular type (binary, continuous, or categorical).
Available parameters will show up after clicking the “Select Parameters…”
button.
Using continuous attribute as an example, attribute file name must be specified,
and parameters and their values can then be selected. The attribute file format
are the same as in PNet, where attribute names are separated using (,) as the
first line, then the attributes listed in space, or tab separated columns. Attribute
file format examples are listed in Appendix A.
Since two sets of actors are involved in BPNet, separate attribute files are
required for parameters involving only one set of actors, either A or P. For
interaction actor attribute effects, the attribute file should list attributes for nodes
- 15 -
in set A first, then followed by attributes for nodes in set P. Putting them in the
other order will produce wrong modeling results.
Simulation options are similar to PNet,
where we can fix the graph density, or use
structural zero files to fix part of the
network and treat them as exogenous.
Sample graphs, degree distributions, and
clustering coefficients can be collected in
separate output files.
Burn in, number of iterations, and number
of samples to pick up are the same as in
PNet.
- 16 -
Estimation
The Network File is text file with a binary rectangular matrix. The number of rows
for the matrix should be the same as the number of Actors(A), and the number of
columns is the number of Actors(P). Note: putting the number of actors in the
other order will produce wrong modeling results.
Other settings for estimation are the same as in PNet.
Goodness of Fit
Goodness of fit settings are the same as in simulation, except the network file
needs to be specified in the same way as in estimation.
- 17 -
References
A. Baddeley and J. Möller. Nearest-neighbour markov point processes and
random sets. International Statistical Review, 57:89–121, 1989.
Peter Bore, Mark Hujsman, Tom A.B. Snijders, Christian Steglich, Lotte
Wichers and Evelien Zeggelink. StOCNET: An open software system for the
advanced statistical analysis of social networks. Groningen ICS /
SciencePlus. http://stat.gamma.rug.nl/stocnet/. 2003
P. Erdös and A. Renyi. On the evolution of random graphs. Publications of
the Mathematical Institute of the Hungarian Academy of Science., 5:17–61,
1960.
Ove Frank and David Strauss. Markov graphs. Journal of the American
Statistical Association, 81:832–842, Sep. 1986.
Charles J. Geyer and Elizabeth A. Thompson. Constrained Monte Carlo
maximum likelihood for dependent data. Journal of the Royal Statistical
Society. Series B (Methodological), 54(3):657–699, 1992.
Steven M. Goodreau. Advances in exponential random graph (p*) models
applied to a large social network. Social Networks (Special Edition).,
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Mark S. Handcock. Assessing degeneracy in statistical models of social
Networks, working paper no. 39., Center for Statistics and the Social
Sciences, University of Washington, 2003.
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and Martina Morris, statnet: An R package for the Statistical Modeling of
Social Networks. Funding support from NIH grants R01DA012831 and
R01HD041877. URL http://www.csde.washington.edu/statnet. 2003.
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distributions for directed graphs. Journal of the American Statistical
Association, 76(373):33–50, Mar. 1981.
David R. Hunter. Curved exponential family models for social networks.
Social Networks (Special Edition)., 29:216–230, 2007.
- 18 -
David R. Hunter, Steven M. Goodreau, and Mark S. Handcock. Goodness
of fit of social network models. Journal of the American Statistical
Association, In Press.
Philippa E. Pattison and Garry L. Robins. Neighborhood-based models for
social networks. Social Methodology, 32:301–337, 2002.
Philippa E. Pattison and Garry L. Robins. Building models for social space:
Neighbourhood-based models for social networks and affiliation structures.
Mathematics and Social Sciences, 42(168):11–29, 2004.
Philippa E. Pattison and Stanley Wasserman. Logit models and logistic
regression for social networks, ii. multivariate relations. Brithish Journal of
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Herbert Robbins and Sutton Monro. A stochastic approximation method.
The Annals of Mathematical Statistics, 22(3):400–407, Sep. 1951.
Garry L. Robins and Philippa E. Pattison. Models and Methods in Social
Network Analysis. Interdependencies and Social Processes: Generalized
Dependence Structures. Cambridge University Press, 2005.
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introduction to exponential random graph (p*) models for social networks.
Social Networks (Special Edition)., 29:173–191, 2007.
Garry L. Robins, Philippa E. Pattison, and Stanley Wasserman. Logit
models and logistic regressio for social networks, iii. valued relations.
Psychometrika, 64(3):371–394, Sep. 1999.
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worlds: Global network structures from local processes. American Journal
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Philippa E. Pattison. Recent developments in exponential random graph
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Sociological Methodology, 29:253–280, 1999.
- 19 -
Tom A.B. Snijders. Markov Chain Monte Carlo estimation of exponential
random graph models. Journal of Social Structure, 3:2, 2002.
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Christian Steglich. Siena: Simulation investigation for empirical network
analysis. 2001.
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Handcock. New specifications for exponential random graph models.
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Separating selection from influence. In Press.
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Cambridge University Press, 1994.
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regression for social networks, i. an introduction to markov graphs and p*.
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Network Analysis. An Introduction to Random Graphs, Dependence
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- 20 -
Appendix A – Sample Files
Sample Input Files
Sample network or dyadic
attribute file:
Sample structural zero file:
The file contains a binary matrix
where „1‟ indicates changeable ties,
and „0‟ indicates fixed ties. Applying
this structural zero file example will
fix all the tie variables related to
node 2 and 5. Also ties between
node 1 and 13, node 1 and 14, are
also fixed.
Network files or dyadic attribute
setting files contain the observed or
covariate network of interest in the
adjacency matrix format.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
1
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
1
0
1
1
1
1
1
1
1
1
1
- 21 -
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
1
1
1
1
1
1
1
1
1
1
0
1
0
0
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
0
0
1
1
1
1
1
1
1
1
1
0
1
1
0
1
0
1
1
1
1
1
1
1
1
0
1
1
0
1
1
0
1
1
1
1
1
1
1
0
1
1
0
1
1
1
0
1
1
1
1
1
1
0
1
1
0
1
1
1
1
0
1
1
1
1
1
0
1
1
0
1
1
1
1
1
0
1
1
1
1
0
1
1
0
1
1
1
1
1
1
0
1
1
0
0
1
1
0
1
1
1
1
1
1
1
0
1
0
0
1
1
0
1
1
1
1
1
1
1
1
0
Attribute file formate



Each column represents an attribute.
Each row corresponds to the same row as in the adjacency matrix
Attribute names should be listed in the first line, delimited by „,‟s.
o Note that attribute names should not start with numbers to meet the
SPSS script requirements for variable names.
Sample binary actor attribute
file:
Sample categorical actor
attribute file:
department,club
1
1
3
2
2
3
3
2
1
3
2
1
1
2
2
3
3
1
3
3
2
2
3
2
1
1
1
2
member,gender
1
1
1
1
0
1
1
0
1
1
0
0
1
0
0
0
1
1
1
0
0
1
0
0
0
1
1
0
Sample continuous actor
attribute file:
income,age,performance
1.0
23
2
1.1
34
6
1.1
42
5
0.5
23
4
0.3
24
1
1.1
19
1
1.5
38
2
0.2
49
1
0.1
58
1
0.2
47
2
1.0
24
3
0.2
36
2
0.1
19
4
0.5
20
3
- 22 -
Sample Output Files
 Estimation and goodness of fit output files are tab delimited to
easy creating tables in excel. Following are examples of output
files, and excel tables where applicable.
“start_statistics_[session name].txt”, “end_statistics_[session
name].txt” and “sample_statistics_[session name].txt”
vertices 14
matrix
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 1 1
0 0 0 1 0 0
0 0 1 0 1 0
0 0 0 0 1 0
0 0 0 0 1 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
1
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
***This graph contains:****
vertices
14
arc
24
reciprocity 2
AinS(2.00) 17.62500
AoutS(2.00) 15.25000
AT-T(2.00) 6.50000
A2P-T(2.00) 40.50000
member_interaction
6
gender_interaction
2
member_sender
11
gender_sender
7
member_receiver
13
gender_receiver
12
Digraph Density = 0.13187
In-degree Distribution:(range[0..n-1])
4 2 4 3 0 1 0 0 0 0 0 0 0 0
Standard deviation of in-degree distribution = 1.435697
Skewness of in-degree distribution = 0.508356
Out-degree Distribution:(range[0..n-1])
2 6 1 4 1 0 0 0 0 0 0 0 0 0
Standard deviation of out-degree distribution = 1.220572
Skewness of out-degree distribution = 0.322264
Corr. Coef. between in and out degree distributions = 0.238744
- 23 -
Mean degree = 1.71429
Global Clustering Coefficients:
Cto = 0.18421
Cti = 0.15217
Ctm = 0.16279
Ccm = 0.20930
AKC-T = 0.16049
AKC-D = 0.20000
AKC-U = 0.13953
AKC-C = 0.20988
Geodesic Distribution:(range[1..n-1,inf])
Note: geodesic = shortest path between two nodes.
The geodesic distribution is not based on semi-paths.
24 32 29 20 5 0 0 0 0 0 0 0 0 72
Quartiles of the geodesic distribution.
Note: Quartiles equal to the number of nodes refer to infinite
geodesics.
2 4 14 14
Triad
300
210
120C
120D
120U
201
111D
111U
030T
030C
102
021D
021C
021U
012
003
Census:
0
0
1
0
2
0
6
2
2
2
13
10
20
12
130
164
“simulation_[session name].txt”
id arc recip in2star out2star in3star out3star uktri
1000 16 3 6 5 0 0 0.00
2000 12 1 3 4 0 1 2.00
3000 14 1 5 3 1 0 1.00
4000 14 2 3 2 0 0 0.00
5000 15 2 4 4 0 0 3.00
6000 15 1 4 3 1 0 1.00
…
…
…
97000 18 2 6 7 0 1 3.00
98000 13 1 2 1 0 0 0.00
99000 17 1 4 5 0 0 0.00
100000 19 4 5 6 0 0 1.00
- 24 -
“parameter_[session name].txt”
Simulation result for digraph with 14 of vertices.
Parameter Values of:
arc
-1.50000
reciprocity 1.20000
2-in-star
-1.30000
2-out-star -1.20000
3-in-star
-1.10000
3-out-star -1.30000
AT-U(2.00) 1.50000
Proposed 1000000 digraphs.
Samples are picked up at 1 per 1000 digraphs.
Accepted 127343 proposed digraphs.
“estimation_[session name].txt”
STOCHASTIC ESTIMATION FOR NETWORK example
ESTIMATION SETTINGS
Number of sub-phases in estimation (phase 2) = 5
starting a-value in estimation (phase 2) = 0.010000
Multiplication factor for estimation (phase 2) = 10
Number of steps in final simulation (phase 3) = 500
Number of estimation runs = 10
STOCHASTIC APPROXIMATION RUN 1
original statistics:24.000000 2.000000 17.625000 15.250000 6.500000
40.500000
starting parameters:-2.902123 0.200148 0.546233 -0.013692 -0.046386
0.143345
Phase1 started with the following setup:
a = 0.010000
num of steps = 25
num of iterations in each step = 224.378698
************************************
mean statistics in phase1:22.880000 1.760000 16.556250 14.147500
5.880000 37.625000
END PHASE1 parameter:-2.902123 0.200148 0.546233 -0.013692 -0.046386
0.143345
Phase 2 started
Subphase 0 started with a valued 0.010000
Subphase 0 has gone up to 213 steps
Parameter after Subphase 0:-2.90351 0.33987
0.12749
Subphase 1 started with a valued 0.010000
Subphase 1 has gone up to 233 steps
Parameter after Subphase 1:-2.86694 0.26247
0.12810
- 25 -
0.55254 -0.01341 -0.03369
0.54200 -0.01491 -0.04047
Subphase 2 started with a valued 0.005000
…
Subphase 4 started with a valued 0.001250
Subphase 4 has gone up to 725 steps
Parameter after Subphase 4:-2.90684 0.20529 0.55291 -0.00977 -0.04367
0.14037
END PHASE2 parameter:-2.906837 0.205294 0.552914 -0.009773 -0.043674
0.140375
Phase3 started with the following setup:
num of steps = 500
num of iterations in each step = 224.378698
***************************************
mean statistics in phase3:24.616000 2.096000 18.135141 15.876000
6.834750 42.064875
Estimation Result for Network SUMMARY (parameter, standard error, tstatistics)
NOTE: t-statistics = (observation - sample mean)/standard error
effects
estimates
stderr
t-ratio
arc
-2.906837
0.98133
-0.08919
*
reciprocity 0.205294
0.81794
-0.05820
AinS(2.00) 0.552914
0.52603
-0.06192
AoutS(2.00) -0.009773
0.59406
-0.07868
AT-T(2.00) -0.043674
0.51150
-0.06069
A2P-T(2.00) 0.140375
0.19825
-0.07173
Estimated Covariance Matrix
0.963005
-0.009254
-0.318091
-0.009254
0.669034
-0.027916
-0.318091
-0.027916
0.276712
-0.384405
-0.058325
0.073059
0.264326
0.007124
-0.151908
-0.107297
-0.001399
0.009170
effects
arc
reciprocity
AinS(2.00)
AoutS(2.00)
AT-T(2.00)
A2P-T(2.00)
estimates
-2.906837
0.205294
0.552914
-0.009773
-0.043674
0.140375
stderr
0.98133
0.81794
0.52603
0.59406
0.5115
0.19825
-0.384405
-0.058325
0.073059
0.352912
-0.136727
0.002928
t-ratio
-0.08919
-0.0582
-0.06192
-0.07868
-0.06069
-0.07173
0.264326
0.007124
-0.151908
-0.136727
0.261629
-0.038435
-0.107297
-0.001399
0.009170
0.002928
-0.038435
0.039304
0.264326
0.007124
-0.151908
-0.136727
0.261629
-0.038435
-0.107297
-0.001399
0.009170
0.002928
-0.038435
0.039304
*
“covariance_[session name].txt”
0.963005
-0.009254
-0.318091
-0.384405
0.264326
-0.107297
-0.009254
0.669034
-0.027916
-0.058325
0.007124
-0.001399
-0.318091
-0.027916
0.276712
0.073059
-0.151908
0.009170
-0.384405
-0.058325
0.073059
0.352912
-0.136727
0.002928
- 26 -
“gof_[session name].txt”
GOODNESS OF FIT
Parameter Values:
arc
-1.19735
reciprocity 0.28248
2-in-star
0.00000
…
Isolates
0.00000
AinS(2.00) 0.61944
AoutS(2.00) -0.92446
AinS(2.00) 0.00000
…
K-L-star(2.00)
0.00000
AT-T(2.00) -0.02294
…
AT-TDU(2.00)
0.00000
A2P-T(2.00) 0.21751
A2P-D(2.00) 0.00000
…
A2P-TDU(2.00)
0.00000
member_interaction
0.39270
gender_interaction
-1.01727
member_sender
-1.12695
gender_sender
-1.33001
member_receiver
-0.18683
gender_receiver
0.49272
…
gender_out2star
0.00000
Simulated 1000000 digraphs.
Statistic samples are picked up at 1 per 1000 digraphs.
Accepted 262407 proposed digraphs.
observation, sample mean (standard error), t-statistic
t-statistics = (observation - sample mean)/standard deviation
effects
arc
24
reciprocity
2-in-star
2-out-star
3-in-star
3-out-star
path2 43
T1
0
T2
0
T3
1
T4
0
T5
2
T6
0
T7
7
T8
7
T9(030T)
T10(030C)
Sink 0
observed
mean stddev
t-ratio
24.135
4.888 -0.028
2
2.063 1.467 -0.043
23
23.997
9.764 -0.102
19
20.436
9.849 -0.146
13
15.866
12.423
-0.231
8
12.216
11.910
-0.354
43.605
18.853
-0.032
0.014 0.133 -0.105
0.357 1.109 -0.322
1.535 2.261 -0.237
0.819 1.266 -0.647
0.687 1.231 1.067
0.918 1.694 -0.542
9.042 8.190 -0.249
7.819 7.847 -0.104
7
7.216 5.313 -0.041
3
2.416 2.082 0.280
1.430 1.101 -1.299
- 27 -
Source
2
2.742 1.311 -0.566
Isolates
2
0.685 0.824 1.596
AinS(2.00) 17.625
17.646
5.961 -0.004
AoutS(2.00) 15.250
15.500
6.049 -0.041
AinS(2.00) 17.625
17.646
5.961 -0.004
AoutS(2.00) 15.250
15.500
6.049 -0.041
K-1-star(2.00)
30.125
27.346
9.801 0.284
1-L-star(2.00)
30.500
28.742
9.674 0.182
K-L-star(2.00)
20.750
18.188
5.091 0.503
AT-T(2.00) 6.500 6.737 4.665 -0.051
AT-C(2.00) 8.500 6.736 5.469 0.323
AT-D(2.00) 7.000 6.656 4.608 0.075
AT-U(2.00) 6.000 6.678 4.577 -0.148
AT-TD(2.00) 6.750 6.697 4.627 0.012
AT-TU(2.00) 6.250 6.708 4.612 -0.099
AT-DU(2.00) 6.500 6.667 4.579 -0.036
AT-TDU(2.00)
6.500 6.690 4.602 -0.041
A2P-T(2.00) 40.500
41.202
16.830
-0.042
A2P-D(2.00) 17.500
18.975
8.679 -0.170
A2P-U(2.00) 21.500
22.538
8.644 -0.120
A2P-TD(2.00)
29.000
30.089
12.376
-0.088
A2P-TU(2.00)
31.000
31.870
12.187
-0.071
A2P-DU(2.00)
19.500
20.756
7.982 -0.157
A2P-TDU(2.00)
26.500
27.572
10.689
-0.100
member_interaction
6
5.895 2.458 0.043
gender_interaction
2
1.993 1.417 0.005
member_sender
11
10.940
2.902 0.021
gender_sender
7
7.055 2.243 -0.025
member_receiver
13
12.828
3.821 0.045
gender_receiver
12
12.135
3.445 -0.039
member_interaction_reciprocity
1
0.393 0.655 0.927
gender_interaction_reciprocity
0
0.060 0.242 -0.248
member_activity_reciprocity
2
1.295 1.139 0.619
gender_activity_reciprocity
1
1.346 1.107 -0.313
member_in2star
15
11.970
7.073 0.428
gender_in2star
16
11.759
6.633 0.639
member_path2
18
17.830
10.092
0.017
gender_path2
13
12.390
6.845 0.089
member_out2star
7
6.453 4.351 0.126
gender_out2star
3
2.338 2.058 0.322
Std Dev in-degree dist 1.436 1.408 0.275 0.100
Skew in-degree dist
0.508 0.555 0.490 -0.094
Std Dev out-degree dist 1.221 1.215 0.270 0.019
Skew out-degree dist
0.322 0.600 0.528 -0.526
CorrCoef in-out-degree dists 0.239 0.163 0.293 0.258
Global Clustering Cto
0.184 0.166 0.079 0.233
Global Clustering Cti
0.152 0.138 0.066 0.219
Global Clustering Ctm
0.163 0.152 0.072 0.144
Global Clustering Ccm
0.209 0.147 0.096 0.643
Global Clustering AKC-T 0.160 0.152 0.070 0.124
Global Clustering AKC-D 0.200 0.166 0.077 0.441
Global Clustering AKC-U 0.140 0.137 0.064 0.036
Global Clustering AKC-C 0.210 0.146 0.093 0.683
ACCEPTANCE RATE: 0.2624
SAMPLE GEODESIC DISTRIBUTION
- 28 -
Note: geodesic = shortest path between two nodes.
The geodesic distribution is not based on semi-paths.
FIRST QUARTILES
Median of sample G25s: 2
Interquartile range: 1
Observed first quartile geodesic: 2
in model samples, 0.00% of graphs have lower G25.
in model samples, 27.50% of graphs have higher G25.
SECOND QUARTILES
Median of sample G50s: 4
Interquartile range: 11
Observed median geodesic: 4
in model samples, 36.80% of graphs have lower G50.
in model samples, 48.40% of graphs have higher G50.
THIRD QUARTILES
Median of sample G75s: 14
Interquartile range: 0
Observed first quartile geodesic: 14
in model samples, 14.00% of graphs have lower G75.
in model samples, 0.00% of graphs have higher G75.
GOF on Triad Census
Triad observed
mean stddev
t-ratio
300
0
0.014 0.133 -0.105
210
0
0.273 0.636 -0.429
120C 1
0.905 1.138 0.083
120D 0
0.504 0.800 -0.630
120U 2
0.372 0.693 2.350
201
0
0.603 1.198 -0.503
111D 6
5.020 3.855 0.254
111U 2
4.061 3.415 -0.603
030T 2
3.656 2.788 -0.594
030C 2
1.210 1.327 0.596
102
13
12.100
8.607 0.105
021D 10
9.375 4.749 0.132
021C 20
20.389
8.096 -0.048
021U 12
11.845
4.778 0.032
012
130
129.376
16.257
0.038
003
164
164.297
29.134
-0.010
Mahalanobis distance =7.168743 (51.390873)
50% simulated samples have smaller Mahalanobis distances than the
observed network.
effects
arc
reciprocity
2-in-star
2-out-star
3-in-star
observed
24
2
23
19
13
- 29 -
mean
24.135
2.063
23.997
20.436
15.866
stddev
4.888
1.467
9.764
9.849
12.423
t-ratio
-0.028
-0.043
-0.102
-0.146
-0.231
3-out-star
path2
T1
T2
T3
T4
T5
T6
T7
T8
T9(030T)
T10(030C)
Sink
Source
Isolates
AinS(2.00)
AoutS(2.00)
AinS(2.00)
AoutS(2.00)
K-1-star(2.00)
1-L-star(2.00)
K-L-star(2.00)
AT-T(2.00)
AT-C(2.00)
AT-D(2.00)
AT-U(2.00)
AT-TD(2.00)
AT-TU(2.00)
AT-DU(2.00)
AT-TDU(2.00)
A2P-T(2.00)
A2P-D(2.00)
A2P-U(2.00)
A2P-TD(2.00)
A2P-TU(2.00)
A2P-DU(2.00)
A2P-TDU(2.00)
member_interaction
gender_interaction
member_sender
gender_sender
member_receiver
gender_receiver
member_interaction_reciprocity
gender_interaction_reciprocity
member_activity_reciprocity
8
43
0
0
1
0
2
0
7
7
7
3
0
2
2
17.625
15.25
17.625
15.25
30.125
30.5
20.75
6.5
8.5
7
6
6.75
6.25
6.5
6.5
40.5
17.5
21.5
29
31
19.5
26.5
6
2
11
7
13
12
1
0
2
- 30 -
12.216
43.605
0.014
0.357
1.535
0.819
0.687
0.918
9.042
7.819
7.216
2.416
1.43
2.742
0.685
17.646
15.5
17.646
15.5
27.346
28.742
18.188
6.737
6.736
6.656
6.678
6.697
6.708
6.667
6.69
41.202
18.975
22.538
30.089
31.87
20.756
27.572
5.895
1.993
10.94
7.055
12.828
12.135
0.393
0.06
1.295
11.91
18.853
0.133
1.109
2.261
1.266
1.231
1.694
8.19
7.847
5.313
2.082
1.101
1.311
0.824
5.961
6.049
5.961
6.049
9.801
9.674
5.091
4.665
5.469
4.608
4.577
4.627
4.612
4.579
4.602
16.83
8.679
8.644
12.376
12.187
7.982
10.689
2.458
1.417
2.902
2.243
3.821
3.445
0.655
0.242
1.139
-0.354
-0.032
-0.105
-0.322
-0.237
-0.647
1.067
-0.542
-0.249
-0.104
-0.041
0.28
-1.299
-0.566
1.596
-0.004
-0.041
-0.004
-0.041
0.284
0.182
0.503
-0.051
0.323
0.075
-0.148
0.012
-0.099
-0.036
-0.041
-0.042
-0.17
-0.12
-0.088
-0.071
-0.157
-0.1
0.043
0.005
0.021
-0.025
0.045
-0.039
0.927
-0.248
0.619
gender_activity_reciprocity
member_in2star
gender_in2star
member_path2
gender_path2
member_out2star
gender_out2star
Std Dev in-degree dist
Skew in-degree dist
Std Dev out-degree dist
Skew out-degree dist
CorrCoef in-out-degree dists
Global Clustering Cto
Global Clustering Cti
Global Clustering Ctm
Global Clustering Ccm
Global Clustering AKC-T
Global Clustering AKC-D
Global Clustering AKC-U
Global Clustering AKC-C
1
15
16
18
13
7
3
1.436
0.508
1.221
0.322
0.239
0.184
0.152
0.163
0.209
0.16
0.2
0.14
0.21
- 31 -
1.346
11.97
11.759
17.83
12.39
6.453
2.338
1.408
0.555
1.215
0.6
0.163
0.166
0.138
0.152
0.147
0.152
0.166
0.137
0.146
1.107
7.073
6.633
10.092
6.845
4.351
2.058
0.275
0.49
0.27
0.528
0.293
0.079
0.066
0.072
0.096
0.07
0.077
0.064
0.093
-0.313
0.428
0.639
0.017
0.089
0.126
0.322
0.1
-0.094
0.019
-0.526
0.258
0.233
0.219
0.144
0.643
0.124
0.441
0.036
0.683
Appendix B – Model Parameter Description
Non-directed Graphs
Parameters Without Actor Attributes
Edge (L)
Isolate
2-Star (S2)
3-Star (S3)
Triangle (T1)
Alt-Triangle (AT)
Alt-Star (AS)
Alt-2-Path (A2P)
2-Triangle (T2)
Bow-Tie
3-Path
4-Cycle
1-Edge-Triangle
(1-ET)
2-Edge-Triangle
(2-ET)
Alt-Edge-Triangle
(AET)
4-Clique
5-Clique
6-Clique
7-Clique
Alt-Clique (AC)
Parameters with Actor Attributes
– actors with attribute
– actors with or without attribute
[Attr] – attribute name
[Attr]-interaction
[Attr]-activity
[Attr]-T3u
[Attr]-T2u
- 32 -
[Attr]-T1u
[Attr]-O3u
[Attr]-O2au
[Attr]-O1au
[Attr]-O2bu
[Attr]-O1bu
Parameters for Continuous Attributes
[Attr]-Sum
+
[Attr]-interaction
x
[Attr]-difference1
Parameters for Categorical Attributes
[Attr]-Matching
[Attr]-Mismatch
Parameters for Dyadic Attributes
Dyadic covariate
1
[Attr]-Edge
[Attr]-S21
[Attr]-S22
[Attr]-T1
[Attr]-T2
[Attr]-T3
Absolute difference between two actor attributes
- 33 -
-
Directed Graphs
Parameters Without Actor Attributes
Arc
sink
Reciprocity
source
In-2-star
Out-2-star
In-3-star
Out-3-star
2-path
T7
T8
T4
T5
T3
T6
T2
Transitive Triad
(T9)
Cyclic Triad (T10)
T1
isolate
Alt-in-star (AinS)
Alt-out-star
(AoutS)
Alt-in-1-out-star
(Ain1outS)
1-in-alt-out-star
(1inAoutS)
Alt-in-alt-out-star
(AinAoutS)
AT-T
AT-C
AT-D
AT-U
A2P-T
A2P-U
A2P-D
- 34 -
Parameters with Actor Attributes
– actors with attribute
– actors with or without attribute
[Attr] – attribute name
[Attr]-Interaction
[Attr]-Interactionreciprocity
[Attr]-Sendermissing
[Attr]-Receivermissing
[Attr]-Activityreciprocity
[Attr]-in-2-star
[Attr]-2-path
[Attr]-Sender
[Attr]-Receiver
[Attr]-out-2-star
Parameters for Continuous Attributes
[Attr]-Sender
[Attr]-Receiver
[Attr]-Receivermissing
[Attr]-Sender-missing
[Attr]-Sum
+
[Attr]-Difference
-
[Attr]-Product
x
+
[Attr]-Differencereciprocity
-
[Attr]-Sumreciprocity
[Attr]-Productreciprocity
[Attr]-in-2-star
[Attr]-2-path
[Attr]-out-2-star
Parameters for Categorical Attributes
[Attr]-Matching
[Attr]-Mismatch
[Attr]-Mismatchreciprocity
[Attr]-Matching-reciprocity
Parameters for Dyadic Attributes
Dyadic covariate
[Attr]-Arc
- 35 -
x
XPNet Graph Statistics
Parameters for two nondirected networks (A and B)
Network A
Network B
EdgeAB
2-StarAB
3-Star-AAB
3-Star-ABB
TriangleAAB
TriangleABB
Binary Attributes
Rab
Rbab
Continuous Attributes
SumAB
+
DifferenceAB
Categorical attributes
Same-category-AB
Diff- category -AB
- 36 -
-
Parameters for two directed networks (A and B)
Network A
Network B
ArcAB
ReciprocityAB
ReciprocityAAB
ReciprocityABB
ReciprocityAABB
In-2-StarAB
Out-2-StarAB
Mixed-2-StarAB
In-3-Star-AAB
Out-3-Star-AAB
In-3-Star-ABB
Out-3-Star-ABB
T-ABB
T-BAA
T-AAB
T-BBA
T-ABA
T-BAB
- 37 -
C-AAB
C-ABB
Binary Attributes
Mrs
Mrr
Mrb
Mrbm
Mrm
Continuous Attributes
Msum
+
Mdiff
-
Msumm
+
Mdiffm
-
Categorical attributes
Same-cate-arcAB
Diff-cate-arcAB
Same-cate-reciAB
Diff-cate-reciAB
- 38 -
BPNet Graph Statistics
Set-P
Set-A
L
Sp2
Sa2
Sp3
Sa3
L3
C4
K-Sp
K-Sa
K-Cp
K-Ca
Binary Attributes
– actors with attribute
– actors with or without attribute
[Attr] – attribute name
[Attr]_RA
[Attr]_RP
[Attr]_TSCA
[Attr]_TSCP
- 39 -
[Attr]_TSOA1
[Attr]_TSOP1
[Attr]_TSOA2
[Attr]_TSOP2
[Attr]_C4A1
[Attr]_C4P1
[Attr]_C4A2
[Attr]_C4P2
[Attr]_rAP
Continuous Attributes
[Attr]_RAC
[Attr]_RPC
[Attr]_TSCAC
[Attr]_TSCPC
[Attr]_TSOACS
+
[Attr]_TSOPCS
[Attr]_TSOACD
-
[Attr]_TSOPCD
- 40 -
+
-
[Attr]_C4ACS
[Attr]_C4PCS
+
[Attr]_C4ACD
[Attr]_C4PCD
[Attr]_RAPC
Categorical Attributes
[Attr]_2path_match_A
[Attr]_2path_match_P
[Attr]_2path_mismatch_A
[Attr]_2path_mismatch_P
[Attr]_4cycle_match_A
[Attr]_4cycle_match_P
[Attr]_4cycle_mismatch_A
[Attr]_4cycle_mismatch_P
- 41 -
+
IPNet Graph Statistics
Denotes actors with attribute.
Denotes actors with or without attribute.
Attribute Density
Star2
Activity
Star3
Two-PathEquivalence
PartnerResource
Contagion
Partner-Activity
T1
T2
T3
Setting matrix
SettingHomophily
Distance matrix
GeographicHomophily
Contagionamong-partners
Remoteness
Remoteness-topartners
Parameters for Binary Attributes
oOb
o_Ob
Parameters for Continuous Attributes
oOc
o_Oc
- 42 -
Parameters for Categorical Attributes
oO_Osame
oO_Odiff
- 43 -